Machine Learning Logistic Regression

Logistic regression Name is somewhat misleading. Really a technique for classification, not regression. “Regression” comes from fact that we fit a linear model to the feature space. Involves a more probabilistic view of classification.

Different ways of expressing probability Consider a two-outcome probability space, where: p( O1 ) = p p( O2 ) = 1 – p = q Can express probability of O1 as: notation range equivalents standard probability p 0.5 1 odds p / q +  log odds (logit) log( p / q ) - 

Log odds Numeric treatment of outcomes O1 and O2 is equivalent If neither outcome is favored over the other, then log odds = 0. If one outcome is favored with log odds = x, then other outcome is disfavored with log odds = -x. Especially useful in domains where relative probabilities can be miniscule Example: multiple sequence alignment in computational biology

From probability to log odds (and back again)

Standard logistic function

Logistic regression Scenario: A multidimensional feature space (features can be categorical or continuous). Outcome is discrete, not continuous. We’ll focus on case of two classes. It seems plausible that a linear decision boundary (hyperplane) will give good predictive accuracy.

Using a logistic regression model Model consists of a vector  in d-dimensional feature space For a point x in feature space, project it onto  to convert it into a real number z in the range -  to +  Map z to the range 0 to 1 using the logistic function Overall, logistic regression maps a point x in d- dimensional feature space to a value in the range 0 to 1

Using a logistic regression model Can interpret prediction from a logistic regression model as: A probability of class membership A class assignment, by applying threshold to probability threshold represents decision boundary in feature space

Training a logistic regression model Need to optimize  so the model gives the best possible reproduction of training set labels Usually done by numerical approximation of maximum likelihood On really large datasets, may use stochastic gradient descent

Logistic regression in one dimension

Logistic regression in one dimension

Logistic regression in one dimension Parameters control shape and location of sigmoid curve  controls location of midpoint  controls slope of rise

Logistic regression in one dimension

Logistic regression in one dimension

Logistic regression in two dimensions Subset of Fisher iris dataset Two classes First two columns (SL, SW) decision boundary

Logistic regression in two dimensions Interpreting the model vector of coefficients From MATLAB: B = [ 13.0460 -1.9024 -0.4047 ]  = B( 1 ),  = [ 1 2 ] = B( 2 : 3 ) ,  define location and orientation of decision boundary -  is distance of decision boundary from origin decision boundary is perpendicular to  magnitude of  defines gradient of probabilities between 0 and 1 

Heart disease dataset 13 attributes (see heart.docx for details) 2 demographic (age, gender) 11 clinical measures of cardiovascular status and performance 2 classes: absence ( 1 ) or presence ( 2 ) of heart disease 270 samples Dataset taken from UC Irvine Machine Learning Repository: http://archive.ics.uci.edu/ml/datasets/Statlog+(Heart) Preformatted for MATLAB as heart.mat.

MATLAB interlude matlab_demo_05.m

Logistic regression Advantages: Disadvantages: Makes no assumptions about distributions of classes in feature space Easily extended to multiple classes (multinomial regression) Natural probabilistic view of class predictions Quick to train Very fast at classifying unknown records Good accuracy for many simple data sets Resistant to overfitting Can interpret model coefficients as indicators of feature importance Disadvantages: Linear decision boundary